Algebra Examples

$f (x) = x^{4} - 1 x^{2}$

Step 1

Determine if the function is odd, even, or neither in order to find the symmetry.

1. If odd, the function is symmetric about the origin.

2. If even, the function is symmetric about the y-axis.

Step 2

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$f (x) = x^{4} - x^{2}$

Step 3

Find $f (- x)$ .

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Step 3.1

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = {(- x)}^{4} - {(- x)}^{2}$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{4} x^{4} - {(- x)}^{2}$

Step 3.2.2

Raise $- 1$ to the power of $4$ .

$f (- x) = 1 x^{4} - {(- x)}^{2}$

Step 3.2.3

Multiply $x^{4}$ by $1$ .

$f (- x) = x^{4} - {(- x)}^{2}$

Step 3.2.4

Apply the product rule to $- x$ .

$f (- x) = x^{4} - ({(- 1)}^{2} x^{2})$

Step 3.2.5

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.2.5.1

Move ${(- 1)}^{2}$ .

$f (- x) = x^{4} + {(- 1)}^{2} \cdot (- 1 x^{2})$

Step 3.2.5.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 3.2.5.2.1

Raise $- 1$ to the power of $1$ .

$f (- x) = x^{4} + {(- 1)}^{2} \cdot ((- 1) x^{2})$

Step 3.2.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- x) = x^{4} + {(- 1)}^{2 + 1} x^{2}$

Step 3.2.5.3

Add $2$ and $1$ .

$f (- x) = x^{4} + {(- 1)}^{3} x^{2}$

Step 3.2.6

Raise $- 1$ to the power of $3$ .

$f (- x) = x^{4} - x^{2}$

Step 4

A function is even if $f (- x) = f (x)$ .

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Step 4.1

Check if $f (- x) = f (x)$ .

Step 4.2

Since $x^{4} - x^{2} = x^{4} - x^{2}$ , the function is even.

The function is even

Step 5

Since the function is not odd, it is not symmetric about the origin.

No origin symmetry

Step 6

Since the function is even, it is symmetric about the y-axis.

Y-axis symmetry

Step 7